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Derived mapping spaces between little $d$-disks operads $E_d$ play an important role in embedding calculus. For example, Dwyer-Hess expresses the homotopy of framed long knots as loop spaces such mapping spaces, a result which was generalized by Boavida de Brito-Weiss.

When stating these results, the $E_d$-operad is usually unitary: its space of 0-ary operations is contractible (in fact a point). There is also a non-unitary version $E_d^{nu}$, where replace the $0$-ary operations by an empty set. There is a natural forgetful map from the derived mapping space of unitary operads to that of non-unitary operads

$$Map^h_{\mathsf{Op}}(E_m,E_n) \to Map^h_{\mathsf{Op}}(E_m^{nu},E_n^{nu}).$$

Is this a weak equivalence?

This is known if we assume $m=n$ and we replace the operads by their rationalizations, by Section 7 of Fresse-Willwacher.

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    $\begingroup$ It was pointed out to me that arxiv.org/pdf/1111.2723.pdf proves that in the case $m=1$ it is an injection on $\pi_0$ and a weak equivalence onto the path components it hits. $\endgroup$ Commented Nov 26, 2020 at 2:28

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A positive answer is the main theorem of my paper with Krannich and Horel, Two remarks on spaces of maps between operads of little cubes. The proof uses a result of Haugseng and Kock to reduce it to a theorem of Lurie.

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